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In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula…

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time $\tau$. Under minimal assumptions on the random time and on the…

Recent empirical evidence has highlighted the crucial role of jumps in both price and volatility within the cryptocurrency market. In this paper, we integrate price--volatility co-jumps and volatility short-term dependency into a coherent…

Black-Scholes implied volatility is a quantile. The insight follows from the normalized option price being a probability on the variance scale, with the inverse Gaussian distribution providing the link. It enables analytically exact and…

The sub-fractional Brownian motion (sfBm) is a stochastic process, characterized by non-stationarity in their increments and long-range dependency, considered as an intermediate step between the standard Brownian motion (Bm) and the…

This paper develops a comprehensive theoretical framework that imports concepts from stochastic thermodynamics to model price impact and characterize the feasibility of round-trip arbitrage in financial markets. A trading cycle is treated…

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the…

Predicting volatility is important for asset predicting, option pricing and hedging strategies because it cannot be directly observed in the financial market. The Black-Scholes option pricing model is one of the most widely used models by…

We introduce fractional Brownian motion processes (fBm) as an alternative model for the turbulent index of refraction. These processes allow to reconstruct most of the refractive index properties, but they are not differentiable. We…

The purpose of this work is to explore the role that random arbitrage opportunities play in pricing financial derivatives. We use a non-equilibrium model to set up a stochastic portfolio, and for the random arbitrage return, we choose a…

Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic…

We characterize absence of arbitrage with simple trading strategies in a discounted market with a constant bond and several risky assets. We show that if there is a simple arbitrage, then there is a 0-admissible one or an obvious one, that…

This paper studies the pricing problem in which the underlying asset follows a non-Markovian stochastic volatility model. Classical partial differential equation methods face significant challenges in this context, as the option prices…

Usually, in the Black-Scholes pricing theory the volatility is a positive real parameter. Here we explore what happens if it is allowed to be a complex number. The function for pricing a European option with a complex volatility has…

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a…

Standard quantitative models of the stock market predict a log-normal distribution for stock returns (Bachelier 1900, Osborne 1959), but it is recognised (Fama 1965) that empirical data, in comparison with a Gaussian, exhibit leptokurtosis…

Volatility of intra-day stock market indices computed at various time horizons exhibits a scaling behaviour that differs from what would be expected from fractional Brownian motion (fBm). We investigate this anomalous scaling by using…

We deal with some generalizations on a Black--Scholes model arising in financial mathematics. As novelty in this paper, we consider a variable volatility and abstract functional boundary conditions, which allow us to treat a very large…

We consider a stochastic volatility model where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is…

We identify an issue in recent approaches to learning-based control that reformulate systems with uncertain dynamics using a stochastic differential equation. Specifically, we discuss the approximation that replaces a model with fixed but…